Computational methods have become the mainstay of scientific investigation in numerous disciplines, and including electromagnetics. Research in both integral equation and differential equation based methods has grown over the past few decades. In recent years, development in both higher order basis functions (e.g., 2, 3, 4, 5, 6) and higher order representations of geometry has increased.
Despite advances in these areas, there is a fundamental disconnect between processing a geometry of an object and performing an analysis based on the geometry. Traditional analysis proceeds by defining a discrete representation of the geometry typically comprising piecewise continuous tessellations. Ironically, this discrete representation of the geometry is obtained using software or a computer aided design (CAD) tool that contains a higher order differentiable representation of the geometry. The rationale for this disconnect is attributed to the different periods in time that CAD tools and analysis tools developed. As the latter is older, the computational foundation is older as well. As a result, one is left with awkward communication with the CAD software for refining and remeshing. This is especially true insofar as accuracy is concerned; lack of higher order continuity in geometry can cause artifacts if the underlying spaces for field representations are not properly defined. Indeed, the need to define div/curl conforming spaces on tessellations that are only C0 led to development of novel basis sets that meet this criterion.
An alternate approach that has recently been espoused recently is iso-geometric analysis (IGA). In this approach, the basis functions used to present the geometry are the same as those used to represent the underlying functions. As a result, the features of geometry representation, such as higher order continuity or adaptivity, carry over to function representation as well. IGA has been applied to a number of applications that range from structure mechanics to fluid-structure interactions (FSI) to contact problems to flow to shell analysis to acoustics and electromagnetics. In addition to analysis techniques, the power of IGA has been harnessed for design-through-analysis phase in several practical applications.
While it may be possible to modify basis functions used for geometry construction to construct the necessary function space to represent the fields, most CAD tools use bi/tri-variate spline based patches/solids like those based upon Bezier, B-splines, and non-uniform rational B-splines (NURBS). As a result, these basis functions are often used as IGA basis, with the most popular being NURBS. The latter choice is determined by the fact that NURBS is the industry standard for modern CAD systems. Properties such as positively and the fact that it provides a partition of unity make it an excellent candidate for defining function spaces. Finite element methods based on NURBS basis functions that exhibit h- and p-adaptivity have been demonstrated. Unfortunately, the challenge with using NURBS arises from the fact that the resulting shapes are topologically either a disk, a tube, or a torus. As a result, stitching together these patches can result in surfaces that are not watertight. These complexities are exacerbated when the object being meshed is topologically complex or has multiple scales.
Two other geometry processing methodologies are T-splines and subdivision. The former is an extension to NURBS and can handle T-junctions and, hence, greatly reduce the number of the control points in the control mesh. T-splines, especially analysis-ready T-splines, comprise a good candidate for constructing isogeometric analysis.
As opposed to T-splines, subdivision surface has played a significant role in the computer animation industry. Among its many advantages is the ease with which one can represent complex topologies, scalability, inherently multiresolution features, efficiency, and ease of implementation. Furthermore, it converges to a smooth limit surface that is C2 almost everywhere, except at isolated points where it is C1. There are several subdivision schemes; Loop, Catmull-Clark, and Doo-Sabin to name a few. Generally speaking, all of the three of these schemes alluded to above can be used to construct an IGA method. To date, isogeometric analysis based on subdivision surface is not popular. Some work on IGA based on Catmull-Clark can be found where IGA is used to solve PDEs defined on a surface.
While the literature on IGA for differential equations is reasonably widespread across multiple fields, IGA for integral equations (lEs) is still at a nascent stage. As a result, it has recently become the focus of significant attention. Recently, the two-dimensional isogeometric boundary element method has been developed to study two-dimensional electromagnetic analysis.
This section provides background information related to the present disclosure which is not necessarily prior art.